Scaling of hysteresis dispersion in a model spin system

Abstract

We present a calculation of the magnetic hysteresis and its area for a model continuum spin system based on three-dimensional $({\ensuremath{\Phi}}^{2}{)}^{2}$ model with $O(N)$ symmetry in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty},$ under a time-varying magnetic field. The frequency dependence of the hysteresis area $A(f),$ namely, hysteresis dispersion, is investigated in detail, predicting a single-peak profile which grows upwards and shifts rightwards gradually with increasing field amplitude ${H}_{0}.$ We demonstrate that the hysteresis dispersion $A(f)$ over a wide range of ${H}_{0}$ can be scaled by scaling function $W(\ensuremath{\eta})\ensuremath{\propto}{\ensuremath{\tau}}_{1}{A(f,H}_{0}),$ where $\ensuremath{\eta}={\mathrm{log}}_{10}(f{\ensuremath{\tau}}_{1})$ and ${\ensuremath{\tau}}_{1}$ is the unique characteristic time for the spin reverse, as long as ${H}_{0}$ is not very small. The inverse characteristic time ${\ensuremath{\tau}}_{1}^{\ensuremath{-}1}$ shows a linear dependence on amplitude ${H}_{0},$ supported by the well-established empirical relations for ferromagnetic ferrites and ferroelectric solids. This scaling behavior suggests that the hysteresis dispersion can be uniquely described by the characteristic time for the spin reversal once the scaling function is available.